Int. J. Appl. Math. Comput. Sci., 2002, Vol.12, No.4, 523–531
CONTROL OF AN INDUCTION MOTOR USING SLIDING MODE LINEARIZATION E RIK ETIEN∗ , S ÉBASTIEN CAUET L AURENT RAMBAULT, G ÉRARD CHAMPENOIS ∗
Laboratoire d’Automatique et d’Informatique Industrielle, 60 avenue du recteur Pineau, 86022 Poitiers, France e-mail:
[email protected]
Nonlinear control of the squirrel induction motor is designed using sliding mode theory. The developed approach leads to the design of a sliding mode controller in order to linearize the behaviour of an induction motor. The second problem described in the paper is decoupling between two physical outputs: the rotor speed and the rotor flux modulus. The sliding mode tools allow us to separate the control from these two outputs. To take account of parametric variations, a model-based approach is used to improve the robustness of the control law despite these perturbations. Experimental results obtained with a laboratory setup illustrate the good performance of this technique. Keywords: induction motor, sliding mode control, linearization
1. Introduction In variable speed domain, many applications need high performances in terms of torques and accuracy. To obtain high performances, several control methods have been developed in the last few years. Sliding mode theory, stemmed from the variable-structure control family, has been used for the Induction Motor (IM) drive for a long time (Utkin, 1999). Introduced as a relatively easier control design, the sliding mode using switched controls produces chattering phenomena and torque perturbations. Many solutions try to limit those drawbacks by using, e.g., smoothed nonlinearities, but at present the main stream of the sliding mode in IM control is the design of flux observers (Edwards and Spurgeon, 1994). On the other hand, many methods of nonlinear system control have been developed, such as exact inputoutput linearization or backstepping (Chiasson, 1996; Taylor, 1994). It is well known that the use of these methods in practical applications needs adaptive solutions to deal with robustness problems (Marino and Valigi, 1991; 1993; Von Raumer et al., 1993a; 1993b). At first, this paper employs sliding mode theory, well known by speed drive conceptors, in order to linearize IM behaviour. The choice of a particular sliding surface permits to create a link between sliding mode theory and input-output linearization. In the second part, the sliding mode in the tracking problem applied to an IM is considered. In the presence of some particular reference input signal, the tracking yields unstable behaviour. To show that the stability depends on the input signals, in Section 4 a stability anal-
ysis based on the Lyapunov theory is presented. A modelbased approach is proposed to solve the stability problem and to improve the robustness of stabilization (Edwards and Spurgeon, 1998). Section 2 presents an IM model in the concordia frame. The third section develops sliding mode theory and its application to linearization. Section 4 introduces a reference model to improve the robustness of linearization. Some experimental results are presented in Section 5.
2. Description of the Electrical Motor In order to design sliding mode control, we use the IM model with respect to a fixed stator reference frame {α, β} (see Appendix). The main reason behind this choice is the improvement of performances concerning the numerical precision. In fact, by using the fixed stator reference frame, it is not necessary to implement a rotor position measurement (Barbot et al., 1992). The objective is to control the following two physical quantities: the rotor speed ω and the magnitude of the rotor flux |Φ|2 = Ψ2rα + Ψ2rβ . We define a new state space representation: ( x˙ = f (x, t) + B(x, t)u, y = C(x, t)
(1)
E. Etien et al.
524 with xT = [ ω
˙ ]. Here |φ|
|φ| ω˙ h
A11
f (x, t) =
Ls = Lsn (1 + δL), i
A12
x
(10)
(2)
In the following part, the controller is designed in two steps in order to control the output vector y T = [ ω |φ| ]. The first step consists in linearizing the behaviour of the nonlinear system (1). The other improves the robustness of linearization and the tracking problem.
(3)
3. Sliding Mode Linearization
f3 (x) f4 (x)
Msr = Msn (1 + δL).
with " A11 =
0 0
0 0
#
" ,
A12 =
1 0
0 1
# .
3.1. Sliding Mode Theory With no loss of generality, the state vector x can be written as follows: " # x1 x= (4) x2 with
" x1 =
x11 x12
#
x21 x22
#
" =
ω |φ|
#
ω˙ ˙ φ
#
where x(t) ∈ Rn , u(t) ∈ Rm and B(x, t) ∈ Rn×m .
(5)
From the system (11), it is possible to define a set S of the state trajectories x such as S = x(t) |σ(x, t) = 0 , (12)
(6)
where
and " x2 =
" =
In this subsection, sliding mode theory is summarized. The reader is referred to (Edwards and Spurgeon, 1998; Utkin, 1992) for details. Let us consider the nonlinear system ( x˙ = f (x, t) + B(x, t)u, (11) y = C(x, t),
.
Then x˙ 1 = x2 .
T σ(x, t) = σ1 (x, t), . . . , σm (x, t) = 0
We find the following functions: f3 (x) = −µpβx11 x12 − (α + γ)x21 −
µpβx11 x22 µpx11 x12 − , 2αMsr Msr
f4 (x) = (4α2 + 2α2 Msr β)x12 +
(13)
and [·]T denotes the transposed vector. (7)
2αpMsr x11 x21 µ
− (3α + γ)x22 " # 2 2 2α2 Msr x22 + 2αx12 x221 + + 2 . (8) x12 2αMsr µ This choice of the state space representation provides a particular form for the matrix f (x, t), which is divided into two blocks: a linear one and a nonlinear one. This particular form, known as the regular form, can be derived from the classical IM model via mathematical transformations (Edwards and Spurgeon, 1998). In order to take account of the parametric variations, we introduce unknown deviations of the resistance (δR) and the inductance (δL). The IM physical parameters can also be defined as follows: ( Rr = Rrn (1 + δR), Rs = Rsn (1 + δR), (9) Lr = Lrn (1 + δL),
S is called the “sliding surface” and the system is said to be in the sliding mode when the state trajectory x of the controller plant satisfies σ(x(t), t) = 0 at every t ≥ t1 for some t1 . The surfaces are designed so that the state trajectory, restricted to σ(x(t), t) = 0, shows some desired behaviour such as stability or tracking. Commonly, in IM control using sliding mode theory, the surfaces are chosen as functions of the error between the reference input signals and the measured signals (Utkin, 1993). After this step, the objective is to determine a control law which drives the state trajectories along the surface (12). The following part shows how to use sliding mode theory for linearization behaviour. In most applications this technique is implemented by using switched controllers in order to improve some performances. The chattering phenomena and the torque perturbations are reduced by adding and designing switch components with hysteresis. The main contribution of this work concerns the application of the sliding mode to linearize a nonlinear system. The next section outlines a method to tune the parameters of the sliding mode controller.
Control of an induction motor using sliding mode linearization 3.2. Application to Linearization The objective of the first step of the design procedure is to convert the nonlinear system (1) to a linear one defined by the following state space representation: ( x˙ = Ax + Bv, (14) y = Cx, where
0 0 A= −s1 s01 0 B=
0 0 1 0
0 0 0 1
0 0 0 −s2 s02
1 0 0 1 (s1 + s01 ) 0 0 (s2 + s02 )
,
T
v =
h
v1
with σ1 = [ S1 S2 ] and σ2 = [ S3 S4 ] standing for matrices to be determined. Developing (18), we can write σ(x, t) = S1 x1 + S2 x2 + S3 x˙ 1 + S4 x˙ 2 . Using (6) yields ¨1 ] . x2 = −(S2 + S3 )−1 [S1 x1 + S4 x
x ¨1 = −S4−1 S1 x1 − S4−1 (S2 + S3 )x˙ 1 .
,
C=
1 0
0 1
0 0
0 0
# ,
h
|φ| ω˙
ω
v2
i
,
T
y =
i
˙ |φ| h
(15)
ω
, |φ|
i
(20)
Substituting the nonlinear regular expression of (3) gives x˙ 1 = −A12 (S2 + S3 )−1 (S1 x1 + S4 x ¨1 ) . (21)
"
(19)
Finally,
with xT =
525
(16)
(22)
Consider the particular case where S4 = S2 = I, " # s1 s01 0 S1 = , 0 s2 s02 (23) " # 0 −(1 + s1 + s1 ) 0 S3 = . 0 −(1 + s2 + s02 )
. Consequently, the sliding surface is
The linearized system is equivalent to two independent subsystems. The first one represents a transfer from the new input, v1 , to the speed of the rotor. The second one is also a linear transfer from an input v2 to the rotor flux modulus. Each subsystem is characterized by two poles: {s1 , s01 } for the first one and {s2 , s02 } for the second one. Thus the linear system can be written using the transfer matrix " # G1 (s) 0 G(s) = 0 G2 (s)
σ(x, t) "
# s1 s01 0 1 0 = x 0 s2 s02 0 1 " # −(1 + s1 + s01 ) 0 1 0 + x. ˙ (24) 0 −(1 + s2 + s02 ) 0 1
In order to find the control law u(t) which imposes σ(x, t) = 0, we use the equivalent control method (Utkin, 1992). Using (18) and (11), we write σ(x, t) = σ1 x + σ2 f (x, t) + σ2 B(x, t)ueq = 0. (25)
with G1 (s) =
ω 1 = , v1 (s − s1 )(s − s01 )
|φ| 1 G2 (s) = = . v1 (s − s2 )(s − s02 ) {s1 , s01 }
(17)
{s2 , s02 }
The eigenvalues and are chosen to determine respectively the dynamics of the rotor speed and the rotor flux modulus in order to consider the physical time constants of the system.
Suppose that the matrix σ2 is such that the square matrix [σ2 B(x, t)] is invertible. Then the equivalent control is defined by the following expression: −1 ueq (x, t) = − σ2 B(x, t) σ1 x + σ2 f (x, t) .
(26)
Finally, the linearizing control is given by −1 u = ueq + σ2 B(x, t) v.
(27)
Consider the particular sliding surface σ(x, t) = σ1 x + σ2 x˙
(18)
The equivalent system with the input vector v is also defined by (14) and its representation is shown in Fig. 1.
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526
Fig. 1. Sliding mode linearization principle.
Notice that this control is equivalent to a classical exact input-output linearization technique with poles placement (Chiasson, 1993; Isidori, 1989). In practice, it is impossible to use directly this type of linearization. The first difficulty is the loss of decoupling when parametric variations appear. This robustness problem is usually solved using an adaptive solution. Another difficulty concerns linearization stability toward the perturbation signal. This stability degree is conditioned by the choice of the pole placement {s1 , s01 }, {s2 , s02 } and the external signals (tracking references, perturbation signals, etc.). In order to improve the limits of linearization, we propose to use the Lyapunov theory. 3.3. Linearization Stability The Lyapunov approach is used for deriving the condition control u(x, t) that will drive the state trajectory to the sliding surface. Consider the quadratic Lyapunov function V (t, x, σ) = σ T (x, t)σ(x, t),
∇σ 6= 0,
δσ δσ δσ B(x, t)u. + f (x, t) + δx δt δx
4. Sliding Mode Control In the presence of parametric variations, the linearized system is modified as follows (Cauet et al., 2001a): x˙ = (An + AR δR + AL δL)x + (Bn + BL δL)v + d(x)
An =
AR = ∇σ 6= 0.
(32)
(34)
with
δσ δσ V˙ = 2σ T + 2σ T f (x, t) δt δx δσ B(x, t)u < 0, δx
(33)
Notice that the result (33) was obtained using the equivalent control (26), which depends on the nominal system (1). In the case of parametric variations or modelling errors, the Lyapunov condition might not be checked. Consequently, for tracking behaviours, the discussed linearization technique cannot be used alone.
(31)
The expression (29) can be written down as follows:
+ 2σ T
∀σ 6= 0.
We see that the Lyapunov condition depends on the pole placement and, consequently, on the designed surface. Moreover, the reference signal may exert influence on linearization stability. Hence, the tracking references are considered as perturbation signals.
(29)
where we have cancelled the specific x and t dependencies. We write δσ δσ (30) σ˙ = + x. ˙ δt δx Equation (1) yields σ˙ =
V˙ = 2σ T v < 0,
(28)
with σ(x, t) being the sliding surface defined in (25). To prove linearization stability, we must verify that the control law (27) allows us to obtain the Lyapunov condition V˙ (t, x, σ) = 2σ T σ˙ < 0,
Equations (27) and (32) yield
0 −s1 s01 0 0
1 s1 + s01 0 0
0 0 0 −γn − αn 0 0 0 0
0 0 0 −s2 s02 0 0 0 AR4,3
0 0 1 s2 + s02
,
0 0 1 −γn − 3αn
AR4,3 = 2αn (−γn − αn + αn βn Msrn ),
,
Control of an induction motor using sliding mode linearization AL =
0 s1 s01 0 0
0 −s1 s01 0 0
0 0 2αn AL4,3
0 0 0
Define the error state
,
e(t) = x(t) − ω(t).
AL4,4 = −2αn − (s2 + s02 ), 0 1 Bn = 0 0 d(x) =
0 0 0 1
,
BL =
0 0 −1 0 0 0 0 −1
0
with
,
0 −δL pµ x11 x22 d1 (x) = + x11 x21 Msrn 2αn
S=
,
d2 (x) with d2 (x) = 2pαn Msrn δL
x11 x12 µ
2 + 2αn2 Msrn (δR + δL ) ( " #) 2 1 x22 + 2αn x21 x212 + 2 . × x21 2αn Msrn µ
We propose to use the particular robust properties of sliding mode control to minimize the consequences of these parametric variations. Commonly, in order to introduce tracking requirements, a model-based approach is chosen. A similar approach is proposed in (Cauet et al., 2001b) for induction motor control. Assume that the plant is defined by (14), which produces the following tracking model: ω˙ = Am ω + Bm r.
(36)
The objective of the second step of the design procedure is to choose the control input vector v which imposes e(t) −→ 0 in a short time. The following sliding surface is proposed: σe (e, t) = Se(t) = 0 (37)
AL4,4
AL4,3 = s2 s02 − 2αn2 βn Msrn − 4αn2 ,
527
(35)
1 0
0 − 1
1 seω 0
0 −
1
.
(38)
seΦ
We have −1/seω and −1/seΦ as the two poles that impose the dynamic errors. The equivalent control v eq (t) (Utkin, 1992) that satisfies the condition σe (e, t) = 0 is v eq (t) = −(SB)−1 S Am e + (A − Am )x − Bm r . (39) To complete the control design, we have to solve the reachability problem (Edwards and Spurgeon, 1998). In the presence of parametric variations, the control v eq (t) cannot impose the condition σe (e, t) = 0. The solution is to find complementary control v N (σe , t) which drives the state trajectory error to the equilibrium manifold. Using the Lyapunov theory, we choose the continuous control (DeCarlo et al., 1996): v N = −P σe (e, t),
P = P T > 0.
(40)
In this case, the derivative of the Lyapunov function can be written as follows: V˙ = 2σeT (e, t)v N = 2σeT (e, t)P σe (e, t).
(41)
The Lyapunov condition V˙ < 0, ∀σe (e, t) 6= 0 is satisfied for all P = P T > 0. Finally, we obtain the global control v(t) = v eq (t) + (SB)−1 v N (t). (42) The control scheme is summarized in Fig. 2.
Fig. 2. Model-based sliding mode control.
E. Etien et al.
528 The reference model imposes the dynamics to be followed by the linearized system. Sliding mode control guarantees the convergence of the error (e(t) −→ 0). The reference model is chosen as a copy of the theorical linearized system in the nominal case. The external loop leads to robust linearization.
5. Experimental Results The control law is tested on an experimental plant composed of a 1.1 kW induction motor. The control command is designed with a digital signal processor TMS320C32 board. The rotor speed and two stator current measurements are used to estimate the error flux component (Verghese and Sanders, 1988). The two poles of the linearized system are s1 = −10, s01 = −200, s2 = s02 = −300. Figure 3 shows the benchmark signals used in practice. To show the advantages of the model-based con-
Fig. 4. Rotor flux response without the RM.
Fig. 3. Benchmark test: rotor speed (turns/min) and load torque (Nm). Fig. 5. Zoom of the rotor flux response without the RM.
trol, we introduce parameter deviations in the control algorithm (δRr = ±50% and δLs = ±20%). The results are provided in two cases: Figs. 4–7 provide the responses obtained without the reference model and Figs. 8–11 illustrate the improvements due to the reference model (RM). Figures 12 and 13 show the sliding surfaces. Without the reference model, the parametric variations introduced in the control algorithm provide an important coupling action between the rotor flux and the rotor speed. Moreover, the speed is obtained with an important steady-state error. With the use of the reference model, the decoupling is ensured and correct tracking is obtained for the components |Φ|2 = Ψ2rα + Ψ2rβ and ω.
6. Conclusion In this paper, we propose induction motor control using sliding mode theory in two steps. A particular sliding sur-
Fig. 6. Rotor speed response without the RM.
Control of an induction motor using sliding mode linearization
529
Fig. 7. Zoom of the rotor speed response without the RM. Fig. 10. Rotor speed response with the RM.
Fig. 8. Rotor flux response with the RM.
Fig. 11. Zoom of the rotor speed response with the RM.
Fig. 9. Zoom of the rotor flux response with the RM.
Fig. 12. Sliding surface on the rotor flux error.
E. Etien et al.
530
Fig. 13. Sliding surface on the rotor speed error.
face allows us to obtain a linearization effect similar to classical linearization techniques. Second feedback based on a reference model is used to obtain robust properties in terms of parameter variations. Experimental results show good performances obtained with this method.
References Barbot J.P., Monaco S., Normand-Cyrot D. and Pantalos N. (1992): Some comments about linearization under sampling. — Proc. Conf. CDC’92, Tucson, Arizona, pp. 2392– 397. Cauet S., Rambault L., Bachelier O. and Mehdi D. (2001a): Robust control and stability analysis of linearized system with parameter variation: Application to an induction motor. — Proc. Conf. CDC’01, Orlando, Floride. Cauet S., Rambault L., Etien E., Mehdi D. and Champenois G. (2001b): Linearizing control of an induction motor subjected to parameter variation. — Rev. Electr. Power Comp. Syst., Vol. 29, No. 7, pp. 629–644. Chiasson J. (1993): Dynamic feedback linearization of the induction motor. — IEEE Trans. Automat. Contr., Vol. 38, pp. 1588–1594. Chiasson J. (1996): Non-linear controllers for an induction motor. — Contr. Eng. Pract., Vol. 4, pp. 977–990. DeCarlo R.A, Zak S.H. and Drakunov S.V. (1996): Variable structure, sliding mode controller design, In: The Control Handbook. — Ch. 57.5, pp. 941–951. Edwards C. and Spurgeon S.K. (1994): On the development of discontinuous observers. — Int. J. Contr., Vol. 25, pp. 1211–1229. Edwards C. and Spurgeon S.K. (1998): Sliding Mode Control, Theory and Application. — Taylor-Francis.
Isidori A. (1989): Nonlinear Control System, 2nd Ed.. — Springer. Marino R. and Valigi P. (1991): Non-linear control of induction motors: A simulation study. — Proc. 1st Europ. Contr. Conf., ECC’91, Grenoble, France, pp. 1057–1062. Marino R., Peresada S. and Valigi P. (1993): Adaptive inputoutput linearizing control of induction motors. — IEEE Trans. Automat. Contr., Vol. 38, pp. 208–221. Taylor D.G. (1994): Non-linear control of electric machines: An overview. — IEEE Contr. Syst. Mag., Vol. 14, pp. 41–51. Utkin V. (1992): Sliding Mode in Control Optimization. — Springer. Utkin V. (1993): Sliding mode control design principles and applications to electric drives. — IEEE Trans. Ind. Electr., Vol. 40, No. 1. Utkin V. (1999): Sliding Mode Control in Electromechanical Systems. — Taylor-Francis. Verghese G. and Sanders S.R. (1988): Observer for flux estimation in induction machines. — IEEE Trans. Ind. Electr., Vol. 35, No. 1, pp. 85–94. Von Raumer T., Dion J.M. and Dugard L. (1993a): Adaptive non-linear control of induction motors with flux observers. — Proc. IEEE Conf. Syst. Man Cybern., SMC’93, Le Touquet, France, pp. 5–84. Von Raumer T., Dion J.M. and Dugard L. (1993b): Adaptive non linear speed and torque control of induction motors. — Int. J. Adapt. Contr. Signal Process., IJACSP, Vol. 7, pp. 435– 455.
Appendix The equations of the motor in the fixed stator reference frame (α, β) are given by z˙ = h(z) + gusαβ ,
(A1)
with z T = [ ω Ψrα Ψrβ isα isβ ], where Ψrα , Ψrβ are the rotor flux dynamics and isα , isβ are the stator currents. The control vector is defined by uTsαβ = [ usα usβ ]. With this notation the state space representation is µ(Ψrα isβ − Ψrβ isα ) − TJl −αΨ − pωΨ + αM i rα rβ sr sα h(z) = pωΨrα − αΨrβ + αMsr isβ , (A2) αβΨrα + pβωΨrβ − γisα −pβωΨrα + αβΨrβ 0 0 0 0 0 0 g= 1 0 σLs 1 0 σLs
− γisβ ,
(A3)
Control of an induction motor using sliding mode linearization and Tl is the load torque. For simplicity, we define the following variables: α=
Rr , Lr
β=
Msr , σLs Lr
M 2 Rr γ = sr 2 + σLs Lr
µ=
pMsr , JLr
531 where Ls is the stator inductance, Msr is the mutual inductance, Lr is the rotor inductance, Rs is the stator resistance, Rr is the rotor resistance and J denotes the rotor inertia.
(A4) Rs σLs ,
Received: 21 May 2001 Revised: 4 July 2002
